I have no idea what the process was for gathering the signatures, but one way to solve this problem is a mutual confidentiality pact that goes away with enough momentum. Ask people if they're willing to sign, but if you can't get 50% of the people to sign the document and all the signatures go away.

Similar things are done when doing anonymous 360 surveys in the workplace. If not enough people in a certain pool respond, the feedback doesn't get shared.